If the level of confidence decreases to 95 but the specified width of the interval

controlled apartments ranged from 200 to 1,500 per month. How many renters are needed in the survey to meet the requirements? 5.17 Refer to Exercise 5.16. Suppose the mayor has reviewed the proposed survey and decides on the following changes:

a. If the level of confidence is increased to 99 with the average rent estimated to

within 25, what sample size is required?

b. Suppose the budget for the project will not support both increasing the level of confi-

dence and reducing the width of the interval. Explain to the mayor the impact on the estimation of the average rent of not raising the level of confidence from 95 to 99. 5.4 A Statistical Test for ␮ 5.18 A researcher designs a study to test the hypotheses H : m ⱖ 28 versus H a : m ⬍ 28. A ran- dom sample of 50 measurements from the population of interest yields ⫽ 25.9 and s ⫽ 5.6.

a. Using a ⫽ .05, what conclusions can you make about the hypotheses based on the

sample information? b. Calculate the probability of making a Type II error if the actual value of m is at most 27. c. Could you have possibly made a Type II error in your decision in part a? Explain your answer.

5.19 Refer to Exercise 5.18. Sketch the power curve for rejecting H : m ⱖ 28 by determining

PWRm a for the following values of m: 22, 23, 24, 25, 26, and 27. a. Interpret the power values displayed in your graph. b. Suppose we keep n ⫽ 50 but change to a ⫽ .01. Without actually recalculating the values for PWRm a , sketch on the same graph as your original power curve, the new power curve for n ⫽ 50 and a ⫽ .01.

c. Suppose we keep a ⫽ .05 but change to n ⫽ 20. Without actually recalculating the

values for PWRm a , sketch on the same graph as your original power curve the new power curve for n ⫽ 20 and a ⫽ .05. 5.20 Use a computer software program to simulate 100 samples of size 25 from a normal distri- bution with m = 30 and s ⫽ 5. Test the hypotheses H : m ⫽ 30 versus H a : using each of the 100 samples of n ⫽ 25 and using a ⫽ .05.

a. How many of the 100 tests of hypotheses resulted in your reaching the decision to

reject H ?

b. Suppose you were to conduct 100 tests of hypotheses and in each of these tests the

true hypothesis was H . On the average, how many of the 100 tests would have re- sulted in your incorrectly rejecting H , if you were using a ⫽ .05? c. What type of error are you making if you incorrectly reject H ? 5.21 Refer to Exercise 5.20. Suppose the population mean was 32 instead of 30. Simulate 100 samples of size n ⫽ 25 from a normal distribution with m ⫽ 32 and s ⫽ 5. Using a ⫽ .05, test the hypotheses H : m ⫽ 30 versus H a : using each of the 100 samples of size n ⫽ 25.

a. What proportion of the 100 tests of hypotheses resulted in the correct decision, that

is, reject H ?

b. In part a, you were estimating the power of the test when m

a ⫽ 32, that is, the ability of the testing procedure to detect that the null hypothesis was false. Now, calculate the power of your test to detect that m ⫽ 32, that is, compute PWRm a ⫽ 32.

c. Based on your calculation in b how many of the 100 tests of hypotheses would you

expect to correctly reject H ? Compare this value with the results from your simulated data. 5.22 Refer to Exercises 5.20 and 5.21. a. Answer the questions posed in these exercises with a ⫽ .01 in place of a ⫽ .05. You can use the data set simulated in Exercise 5.20, but the exact power of the test, PWRm a ⫽ 32, must be recalculated. b. Did decreasing a from .05 to .01 increase or decrease the power of the test? Explain why this change occurred. Med. 5.23 A study was conducted of 90 adult male patients following a new treatment for congestive heart failure. One of the variables measured on the patients was the increase in exercise capacity m ⫽ 30 m ⫽ 30 y